arXiv subjects
Publications and source records attributed to Ronaldo Freire de Lima.
We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.
Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function $W$ of their principal curvatures,called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.
We consider translating solitons to flows by positive powers $α$ of the Gaussian curvature -- called $K^α$-flows -- in Riemannian products $M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb R^n,$ the sphere $\mathbb S^n,$ or one of the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m,$ there exist complete rotational translating solitons to $K^α$-flow in $M\times\mathbb R$ for certain values of $α.$
We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold. If the height function of $f$ has a critical point, then it is an embedding and $M$ is homeomorphic to $\mathbb S^n$ or $\mathbb R^n.$ Furthermore, $f(M)$ bounds a convex set in $\mathscr{H}^n\times\mathbb R.$ In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in $\mathbb S^n\times\mathbb R$ as well. We apply these theorems to show that a compact connected hypersurface in $\mathbb Q_ε^n\times\mathbb R$ ($ε=\pm 1$) is a rotational sphere, provided it has either constant mean curvature and positive-definite second fundamental form or constant sectional curvature greater than $(ε+1)/2.$ We also prove that, for $\bar M=\mathscr H^n$ or $\mathbb S^n,$ any connected proper hypersurface $f:M^n\rightarrow\bar M^n \times\mathbb R$ with positive semi-definite second fundamental form and height function with no critical points is embedded and isometric to $Σ^{n-1}\times\mathbb R,$ where $Σ^{n-1}\subset\bar M^n$ is convex and homeomorphic to $\mathbb S^{n-1}$ (for $\bar M^n=\mathscr H^n$ we assume further that $f$ is cylindrically bounded). Analogous theorems for hypersurfaces in warped product spaces $\mathbb R\times_ρ\mathscr H^n$ and $\mathbb R\times_ρ\mathbb S^n$ are obtained. In all of these results, the manifold $M^n$ is assumed to have dimension $n\ge 3.$